A359040 Sum of the number of divisors of floor(n/(b*c)) with b,c > 0 and b*c <= n.

1, 4, 6, 12, 13, 21, 21, 32, 34, 39, 39, 59, 57, 61, 63, 80, 79, 94, 92, 107, 105, 107, 107, 149, 145, 144, 146, 158, 156, 176, 172, 199, 197, 197, 195, 239, 234, 234, 230, 263, 259, 273, 269, 279, 280, 280, 280, 354, 346, 346, 342, 346, 344

Offset

1,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Jean Bourgain and Nigel Watt, Mean square of zeta function, circle problem and divisor problem revisited, arXiv preprint arXiv:1709.04340 [math.AP], 2017.

Ling Li, On the Hyperbolic Fractional Sum of the Divisor Function, arXiv:2604.20400 [math.NT], 2026.

Ling Li and Meselem Karras, Hyperbolic summation for the divisor function involving the integer part function, arXiv preprint arXiv:2212.05443 [math.NT], 2022.

Formula

Li & Karras prove that a(n) = An log n + Bn + O(n^e) for any e > 10/17, where A and B are defined in their paper (see Links). This sharpens a general result of Bourgain & Watt.

Prog

(PARI) a(n)=2*sum(a=1, n, my(N=n\a); sum(b=1, min(a-1, N), numdiv(N\b))) + sum(a=1, sqrtint(n), numdiv(n\a^2))
(PARI) a(n)=sum(i=1, n, numdiv(i)*numdiv(n\i)) \\ David A. Corneth, Dec 16 2022

Crossrefs

Sequence in context: A310592 A310593 A310594 * A310595 A350864 A175593

Adjacent sequences: A359037 A359038 A359039 * A359041 A359042 A359043

Keyword

nonn

Author

Charles R Greathouse IV, Dec 13 2022

Status

approved